Mercurial > octave
view libinterp/corefcn/schur.cc @ 30564:796f54d4ddbf stable
update Octave Project Developers copyright for the new year
In files that have the "Octave Project Developers" copyright notice,
update for 2021.
In all .txi and .texi files except gpl.txi and gpl.texi in the
doc/liboctave and doc/interpreter directories, change the copyright
to "Octave Project Developers", the same as used for other source
files. Update copyright notices for 2022 (not done since 2019). For
gpl.txi and gpl.texi, change the copyright notice to be "Free Software
Foundation, Inc." and leave the date at 2007 only because this file
only contains the text of the GPL, not anything created by the Octave
Project Developers.
Add Paul Thomas to contributors.in.
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Tue, 28 Dec 2021 18:22:40 -0500 |
parents | a61e1a0f6024 |
children | 08b08b7f05b2 da3ac42a143d |
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//////////////////////////////////////////////////////////////////////// // // Copyright (C) 1996-2022 The Octave Project Developers // // See the file COPYRIGHT.md in the top-level directory of this // distribution or <https://octave.org/copyright/>. // // This file is part of Octave. // // Octave is free software: you can redistribute it and/or modify it // under the terms of the GNU General Public License as published by // the Free Software Foundation, either version 3 of the License, or // (at your option) any later version. // // Octave is distributed in the hope that it will be useful, but // WITHOUT ANY WARRANTY; without even the implied warranty of // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the // GNU General Public License for more details. // // You should have received a copy of the GNU General Public License // along with Octave; see the file COPYING. If not, see // <https://www.gnu.org/licenses/>. // //////////////////////////////////////////////////////////////////////// #if defined (HAVE_CONFIG_H) # include "config.h" #endif #include <string> #include "schur.h" #include "defun.h" #include "error.h" #include "errwarn.h" #include "ovl.h" #include "utils.h" OCTAVE_NAMESPACE_BEGIN template <typename Matrix> static octave_value mark_upper_triangular (const Matrix& a) { octave_value retval = a; octave_idx_type n = a.rows (); assert (a.columns () == n); const typename Matrix::element_type zero = typename Matrix::element_type (); for (octave_idx_type i = 0; i < n; i++) if (a(i, i) == zero) return retval; retval.matrix_type (MatrixType::Upper); return retval; } DEFUN (schur, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {@var{S} =} schur (@var{A}) @deftypefnx {} {@var{S} =} schur (@var{A}, "real") @deftypefnx {} {@var{S} =} schur (@var{A}, "complex") @deftypefnx {} {@var{S} =} schur (@var{A}, @var{opt}) @deftypefnx {} {[@var{U}, @var{S}] =} schur (@dots{}) @cindex Schur decomposition Compute the Schur@tie{}decomposition of @var{A}. The Schur@tie{}decomposition is defined as @tex $$ S = U^T A U $$ @end tex @ifnottex @example @code{@var{S} = @var{U}' * @var{A} * @var{U}} @end example @end ifnottex where @var{U} is a unitary matrix @tex ($U^T U$ is identity) @end tex @ifnottex (@code{@var{U}'* @var{U}} is identity) @end ifnottex and @var{S} is upper triangular. The eigenvalues of @var{A} (and @var{S}) are the diagonal elements of @var{S}. If the matrix @var{A} is real, then the real Schur@tie{}decomposition is computed, in which the matrix @var{U} is orthogonal and @var{S} is block upper triangular with blocks of size at most @tex $2 \times 2$ @end tex @ifnottex @code{2 x 2} @end ifnottex along the diagonal. The diagonal elements of @var{S} (or the eigenvalues of the @tex $2 \times 2$ @end tex @ifnottex @code{2 x 2} @end ifnottex blocks, when appropriate) are the eigenvalues of @var{A} and @var{S}. The default for real matrices is a real Schur@tie{}decomposition. A complex decomposition may be forced by passing the flag @qcode{"complex"}. The eigenvalues are optionally ordered along the diagonal according to the value of @var{opt}. @code{@var{opt} = "a"} indicates that all eigenvalues with negative real parts should be moved to the leading block of @var{S} (used in @code{are}), @code{@var{opt} = "d"} indicates that all eigenvalues with magnitude less than one should be moved to the leading block of @var{S} (used in @code{dare}), and @code{@var{opt} = "u"}, the default, indicates that no ordering of eigenvalues should occur. The leading @var{k} columns of @var{U} always span the @var{A}-invariant subspace corresponding to the @var{k} leading eigenvalues of @var{S}. The Schur@tie{}decomposition is used to compute eigenvalues of a square matrix, and has applications in the solution of algebraic @nospell{Riccati} equations in control (see @code{are} and @code{dare}). @seealso{rsf2csf, ordschur, ordeig, lu, chol, hess, qr, qz, svd} @end deftypefn */) { int nargin = args.length (); if (nargin < 1 || nargin > 2 || nargout > 2) print_usage (); octave_value arg = args(0); std::string ord; if (nargin == 2) ord = args(1).xstring_value ("schur: second argument must be a string"); bool force_complex = false; if (ord == "real") { ord = ""; } else if (ord == "complex") { force_complex = true; ord = ""; } else { char ord_char = (ord.empty () ? 'U' : ord[0]); if (ord_char != 'U' && ord_char != 'A' && ord_char != 'D' && ord_char != 'u' && ord_char != 'a' && ord_char != 'd') { warning ("schur: incorrect ordered schur argument '%s'", ord.c_str ()); return ovl (); } } octave_idx_type nr = arg.rows (); octave_idx_type nc = arg.columns (); if (nr != nc) err_square_matrix_required ("schur", "A"); if (! arg.isnumeric ()) err_wrong_type_arg ("schur", arg); octave_value_list retval; if (arg.is_single_type ()) { if (! force_complex && arg.isreal ()) { FloatMatrix tmp = arg.float_matrix_value (); if (nargout <= 1) { math::schur<FloatMatrix> result (tmp, ord, false); retval = ovl (result.schur_matrix ()); } else { math::schur<FloatMatrix> result (tmp, ord, true); retval = ovl (result.unitary_schur_matrix (), result.schur_matrix ()); } } else { FloatComplexMatrix ctmp = arg.float_complex_matrix_value (); if (nargout <= 1) { math::schur<FloatComplexMatrix> result (ctmp, ord, false); retval = ovl (mark_upper_triangular (result.schur_matrix ())); } else { math::schur<FloatComplexMatrix> result (ctmp, ord, true); retval = ovl (result.unitary_schur_matrix (), mark_upper_triangular (result.schur_matrix ())); } } } else { if (! force_complex && arg.isreal ()) { Matrix tmp = arg.matrix_value (); if (nargout <= 1) { math::schur<Matrix> result (tmp, ord, false); retval = ovl (result.schur_matrix ()); } else { math::schur<Matrix> result (tmp, ord, true); retval = ovl (result.unitary_schur_matrix (), result.schur_matrix ()); } } else { ComplexMatrix ctmp = arg.complex_matrix_value (); if (nargout <= 1) { math::schur<ComplexMatrix> result (ctmp, ord, false); retval = ovl (mark_upper_triangular (result.schur_matrix ())); } else { math::schur<ComplexMatrix> result (ctmp, ord, true); retval = ovl (result.unitary_schur_matrix (), mark_upper_triangular (result.schur_matrix ())); } } } return retval; } /* %!test %! a = [1, 2, 3; 4, 5, 9; 7, 8, 6]; %! [u, s] = schur (a); %! assert (u' * a * u, s, sqrt (eps)); %!test %! a = single ([1, 2, 3; 4, 5, 9; 7, 8, 6]); %! [u, s] = schur (a); %! assert (u' * a * u, s, sqrt (eps ("single"))); %!error schur () %!error schur (1,2,3) %!error [a,b,c] = schur (1) %!error <must be a square matrix> schur ([1, 2, 3; 4, 5, 6]) %!error <wrong type argument 'cell'> schur ({1}) %!warning <incorrect ordered schur argument> schur ([1, 2; 3, 4], "bad_opt"); */ DEFUN (rsf2csf, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {[@var{U}, @var{T}] =} rsf2csf (@var{UR}, @var{TR}) Convert a real, upper quasi-triangular Schur@tie{}form @var{TR} to a complex, upper triangular Schur@tie{}form @var{T}. Note that the following relations hold: @tex $UR \cdot TR \cdot {UR}^T = U T U^{\dagger}$ and $U^{\dagger} U$ is the identity matrix I. @end tex @ifnottex @tcode{@var{UR} * @var{TR} * @var{UR}' = @var{U} * @var{T} * @var{U}'} and @code{@var{U}' * @var{U}} is the identity matrix I. @end ifnottex Note also that @var{U} and @var{T} are not unique. @seealso{schur} @end deftypefn */) { if (args.length () != 2 || nargout > 2) print_usage (); if (! args(0).isnumeric ()) err_wrong_type_arg ("rsf2csf", args(0)); if (! args(1).isnumeric ()) err_wrong_type_arg ("rsf2csf", args(1)); if (args(0).iscomplex () || args(1).iscomplex ()) error ("rsf2csf: UR and TR must be real matrices"); if (args(0).is_single_type () || args(1).is_single_type ()) { FloatMatrix u = args(0).float_matrix_value (); FloatMatrix t = args(1).float_matrix_value (); math::schur<FloatComplexMatrix> cs = math::rsf2csf<FloatComplexMatrix, FloatMatrix> (t, u); return ovl (cs.unitary_schur_matrix (), cs.schur_matrix ()); } else { Matrix u = args(0).matrix_value (); Matrix t = args(1).matrix_value (); math::schur<ComplexMatrix> cs = math::rsf2csf<ComplexMatrix, Matrix> (t, u); return ovl (cs.unitary_schur_matrix (), cs.schur_matrix ()); } } /* %!test %! A = [1, 1, 1, 2; 1, 2, 1, 1; 1, 1, 3, 1; -2, 1, 1, 1]; %! [u, t] = schur (A); %! [U, T] = rsf2csf (u, t); %! assert (norm (u * t * u' - U * T * U'), 0, 1e-12); %! assert (norm (A - U * T * U'), 0, 1e-12); %!test %! A = rand (10); %! [u, t] = schur (A); %! [U, T] = rsf2csf (u, t); %! assert (norm (tril (T, -1)), 0); %! assert (norm (U * U'), 1, 1e-14); %!test %! A = [0, 1;-1, 0]; %! [u, t] = schur (A); %! [U, T] = rsf2csf (u,t); %! assert (U * T * U', A, 1e-14); */ OCTAVE_NAMESPACE_END